1. With an ellipse the foci have the coordinates with

That means the foci are at and

2. A hyperbola is defined by a constant difference of the distances between a point L on the hyperbola and the foci:

with

I've got

3. The excentricity of a hyperbola is calculated by:

Since the excentricity is e = 5 you can determine the value of b.

I've got

(For this part I used my computer)

4. Therefore the equation of the hyperbola is:

5. I've attached a drawing of the ellipse and the hyperbola with the point L on the right branch.